Groups of finite Morley rank
A group of finite Morley rank is a group (G,\cdot) , usually with extra structure, whose Morley rank is less than \omega . The Cherlin-Zilber conjecture asserts that every simple group of finite Morley rank is an algebraic group over a field. This remains open as of 2014. However, a considerable amount is known about groups of finite Morley rank. See for example, Bruno Poizat's book Stable Groups, as well as…recent books * Morley rank and Lascar rank coincide, and are definable. In particular, Morley rank satisfies the Lascar inequalities. * If G is a group of finite Morley rank, then the connected component G^0 exists, and is definable, rather than merely being type-definable. There is a unique type in G^0 of maximal Morley rank, i.e., G^0 has Morley degree 1. The translates of G^0 are called the generics of G , and have many good properties. They are the unique types which are translation invariant. * Any field of finite Morley rank is algebraically closed, but may have additional structure. * A group of finite Morley rank is simple (in the group theoretic sense) if and only if it is definable simple. That is, if G is not simple as an abstract group, then G has a definable normal subgroup. * Every infinite group of finite Morley rank contains an infinite abelian definable subgroup. * Every simple group of finite Morley rank is almost strongly minimal, i.e., is algebraic over a strongly minimal set. * Groups of finite Morley rank are "dimensional." This falls out of the Lascar analysis. * Every type-definable subgroup of a group of finite Morley rank is, in fact, definable. Transitive action on a strongly minimal set One rather strong result about groups of finite Morley rank is the following: Let G be a group of finite Morley rank, acting transitively and faithfully on a strongly minimal set S . Then we are in one of the following three situations: * G has rank 1, is commutative, and S is a G -torsor. * G has rank 2, S is the affine line over a definable field K , and G is the group of affine linear transformations over K * G has rank 3, G is PSL_2(K) for a definable field K , and S is the projective line over K , with the usual action. In cases 2 or 3, K is algebraically closed. G cannot have rank greater than 3. Under the hypothesis that there are no bad groups, it can be shown that this implies that the Cherlin-Zilber conjecture holds for groups of Morley rank at most 3: any simple group of Morley rank at most 3 must be PSL_2(K) for a definable field K . It also implies that if G is a simple group of finite Morley rank, containing a definable subgroup H such that RM(H) = RM(G) - 1 , then G has rank 3 and is PSL_2(K) over an algebraically closed definable field.